Moving Frames for the Numerical Solution of Partial Differential Equations in Complex Domains

This book presents a comprehensive and geometrical approach to solving partial differential equations (PDEs) on complex curved domains using orthonormal moving frames. Rooted in Élie Cartan's classical theory but adapted for computational practicality, the framework aligns local basis vectors with the intrinsic geometry and anisotropy of the domain, enabling accurate and efficient discretization without requiring explicit metric tensors or Christoffel symbols. Topics include the construction of moving frames on general manifolds, covariant derivatives via connection 1-forms, and frame-based formulations of gradient, divergence, curl, and Laplacian operators. Extensive MATLAB and C++ implementations (via Nektar++) are provided for benchmark problems in diffusion-reaction systems, shallow water equations, and Maxwell's equations on complex surfaces such as the sphere, pseudosphere, and atrial tissue. Emphasizing clarity and accessibility, the book blends theory, visualization, and numerical practice, making it an essential resource for graduate students and researchers in scientific computing, applied mathematics, and engineering disciplines dealing with PDEs on non-Euclidean domains.

Unified framework for the numeircal solution of PDEs on curved domains Practical implementation with high-order numerical schemes in MATLAB and C++ Designed for accessibility and visualization of geometric concepts

Autorentext

Sehun Chun is an associate professor of Applied Mathematics at Yonsei University.

Inhalt

Chapter 1 Introduction.- Chapter 2 Constructing moving frames.- Chapter 3 Covariant Derivative in Moving Frames.- Chapter 4 Differential Operators in Moving Frames.- Chapter 5 Applications to PDEs on Curved Surfaces.- Chapter 6 Relative Acceleration and the Riemann Curvature Tensor.